Analytic signals, both continuous and discrete, are signals of-considerable importance in many fields, and especially in signal processing. Instead of processing a given real signal, it is often advantageous to processing its analytic counterpart, which includes in its real part the original signal and in its imaginary part, a 90° phase shifted version of the original signal. The imaginary part is often referred to as the hilbert transform of the real part, and the filter generating the imaginary part is typically referred to as a hilbert transformer or, equivalently, a 90° phase shifter. The analytic signal is, then, a generalization of the complex exponential. Analytic signals are useful in a host of applications in signal processing and in communications. Examples of such applications include, but are not limited to, single-sideband analog communication systems and analog frequency-division multiplex systems.
Discrete-time analytic (DTA) signals have characteristics similar to their analog counterparts. A discrete-time sequence is defined to be analytic by requiring that its discrete-time Fourier transform (DTFT) vanish in the interval [π,0). Such a signal is referred to as a DTA signal. DTA signals may be generated by methods that include frequency-based methods, such as hilbert, and time-domain filtering methods. Time-domain filtering methods are real-time and can convert a real input signal to a DTA signal as the real input signal is received. In a time-domain approach, the length of the filter affects the accuracy and approximation to the analytic signal. Frequency-domain methods are not real-time and require access to the full real input signal. Certain frequency-domain methods, such as hilbert, fail to generate a DTA signal for specific discrete-time real signals and could benefit from improved shiftability. In addition, these methods could benefit from a decreased attenuation of the DTFT in the negative frequency range.
A new frequency-domain method is desired that can generate a DTA signal for those specific discrete-time real signals for which other frequency-domain methods fail. Also, it is desireable to generate a DTA signal for any discrete-time real signal with decreased attenuation of the DTFT in the negative frequency range as measured by the property of shiftability. In addition, in the time domain it is desirable to have a new “uniform approximation” (i.e., an N-length filter having an impulse response that converges uniformly to an ideal response as N→∞) finite impulse response (FIR) filter that filters a discrete real signal and outputs a DTA signal and is characterized by invertibility, real-time implementation and linear phase and, depending upon the application, also orthogonality.